Abstract:
Let $M$ be any metric space with distance $d(x,y)$. Define the function
$A(M,s)$ to be the maximal number of points in $M$ such that all pairwise
distances between points are not less than $s$, where $s>0$. We consider as
$M$ the unit sphere in the $n$-dimensional Euclidean space and the cube with
vertices whose coordinates are $+1$ or $-1$, in the same space. In the case
of the unit sphere it is convenient to replace Euclidean distance by the
angular distance, and in the case of the cube by the number of
coordinates that are different for two given vectors, which is called a
Hamming distance. The first problem is studied by discrete geometry, and
the second one by the coding theory. We will show some similarity
between these two problems. In particular, in the both cases the
function $A(M,s)$ behaves “in a threshold manner,” i.e., if s greater than
the threshold, then $A(M,s)$ does not increase more than linearly with $n$,
and if $s$ is smaller then $A(M,s)$ grows exponentially. The threshold is 90
degrees for the sphere and is $n/2$ for the cube. However, the order of
the exponent is not known, only the upper and lower bounds are known.
Nevertheless these bounds give, in particular, the best upper and lower
asymptotic bounds for the kissing number, as well as the fact that the
sphere packing density of the Euclidean $n$-dimensional space lies
(asymptotically) between $2^{-n}$ and $2^{-0.599n}$.